auseinander
apart
Expresses a spatial separation or distance between two or more things or people.
Die beiden Häuser stehen weit auseinander.
The two houses are far apart.

apart
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apart
Expresses a spatial separation or distance between two or more things or people.
Die beiden Häuser stehen weit auseinander.
The two houses are far apart.

The word 'auseinander' (apart) is made up of 'aus' (out of) and 'einander' (one another). It literally means 'from one another' or 'out of the group' and always expresses some form of separation or distance.
Nach dem Streit gingen die Freunde im Zorn auseinander.
After the argument, the friends angrily went their separate ways.
'Auseinander-' is a common separable prefix for verbs such as 'auseinandernehmen' (to take apart) or 'auseinandergehen' (to go apart / to separate). In a main clause, the prefix stands at the end.
Der Uhrmacher nimmt das alte Uhrwerk vorsichtig auseinander.
The watchmaker carefully takes the old clockwork apart.
The fixed expression 'jemanden/etwas auseinander halten' (to tell someone/something apart) means to be able to distinguish similar people or things. It is very often used in the negative.
Die eineiigen Zwillinge kann man nur schwer auseinander halten.
It is very hard to tell the identical twins apart.
The phrase 'auseinander sein' (to be apart / to have separated) is a common, rather colloquial way to say that a couple has separated or that friends have fallen out.
Ich habe gehört, dass Maria und Paul schon seit Monaten auseinander sind.
I heard that Maria and Paul have been apart for months.
together
'zusammen' and 'auseinander' often form a clear contrast. 'zusammen' means that people or things are together, close to each other, or acting as a unit. 'auseinander' means that they are separated or have distance between them. With people, 'zusammen sein' can also mean being in a romantic relationship. 'Auseinander sein' can then mean that this relationship is over or that people are apart from each other. The simple contrast is: zusammen = together or close, auseinander = separated or distant.

Nach dem Unterricht gingen die Schüler schnell auseinander.
After class, the students quickly went their separate ways.
Am Morgen haben wir zusammen mit der Familie gefrühstückt.
In the morning, we had breakfast together with the family.